Single‐constant simplification of Kubelka‐Munk turbid‐media theory for paint systems—A review

Abstract

For opaque coloration systems, Kubelka-Munk turbid media theory is used commonly to model optical mixing behavior. Most educational publications on the subject use opaque paint systems when describing the two-constant approach and textile systems when describing the singleconstant simplification. Because of the differences in defining concentration for these systems and the corresponding degrees of freedom, the single-constant simplification for paint and textile systems are not identical. The second edition of ''Principles of Color Technology'' showed a numerical example for an opaque paint system modeled using the textile equations. The third edition used the same example but modified the degrees of freedom, a hybrid of the paint and textile approaches. Recent research by Berns and Mohammadi has evaluated the single-constant simplification for modeling artist paints; they have used both the hybrid and paint approaches. Thus, it was of interest to review these different approaches and determine whether these differences have practical importance and whether future printings and editions of Principles of Color Technology should be modified. The three approaches were tested for tints made from a mixture of cobalt blue and titanium white acrylic emulsion artist paints. The differences between the textile and hybrid approaches were inconsequential. The paint approach was superior and its use is recommended for opaque paint systems. The differences in the numerical example from Principles of Color Technology were very small. For future printings of the third edition, the example will remain unchanged. For future editions, including the numerical example remains an open question.

Linear modeling of modern artist paints using a modification of the opaque form of Kubelka‐Munk turbid media theory

“…These algorithms use linear mixing models and the obvious choice for complex subtractive mixing is the opaque form of single‐constant Kubelka‐Munk (1‐KM) turbid‐media theory. Unfortunately, 1‐KM theory is only effective for paint mixtures where scattering is dominated by white colorants such as pastels . With a reduction in white, many pigments' significant scattering leads to errors both in identification and concentration estimation.…”

Dye binary mixture formulation by means of derivative ratio spectra of the Kubelka–Munk function

“…2,4,10 However, other types of colorant formulation approaches, using modified forms of K-M based algorithms are also found in the literature. 12,17 In the application of single-constant K-M theory, the usual known method for the determination of dye concentration in multicomponent mixtures is; first of all, to make a selection of dyes likely to produce the target color, then apply each dye to a given substrate with different concentrations covering the range of working concentrations normally used. The reflectance values of the different dyed samples at the different concentrations are measured using a reflectance spectrophotometer.…”

“…Generally there are two kinds of K-M model based color matching algorithms. [12][13][14][15][16] The singleconstant K-M based algorithm is usually used for color *Correspondence to: A. Shams Nateri (e-mail: a_shams@guilan.ac.ir). matching of textiles and paper samples and another algorithm, based on two-constant K-M theory is applied in paint systems.…”

Modified Spectral Neugebauer Model for Printer Characterization

“…[2] This paper mainly focuses on the former process. Many mathematical models have been employed during color characterization, such as the spectral Neugebauer equations, [3,4] Kubelka-Munk model, [5] polynomial functions based on least-squares, [6,7] artificial neural networks (ANNs), [8,9] 3D interpolation techniques, [10][11][12][13] and so on. Spectral Neugebauer equations are established based on subtractive coloring principle and widely used for color characterization.…”